Definition df-oc 9063

Description: Define orthogonal complement of a subset (usually a subspace) of Hilbert space. The orthogonal complement is the set of all vectors orthogonal to all vectors in the subset. See ocvalt 9092 and chocval 9110 for its value. Textbooks usually denote this unary operation with the symbol ⊥ as a small superscript, although Mittelstaedt uses the symbol as a prefix operation. Here we define a function (prefix operation) ⊥ rather than introducing a new syntactical form. This lets us take advantage of the theorems about functions that we already have proved under set theory. Definition of [Mittelstaedt] p. 9.

Assertion

Ref	Expression
df-oc	⊢ ⊥ = {⟨x, y⟩∣(x ⊆ ℋ ⋀ y = {z ∈ ℋ ∣∀w ∈ x (z ·ih w) = 0})}

Distinct variable group:   x,y,z,w

Detailed syntax breakdown of Definition df-oc

Step	Hyp	Ref	Expression
1		cort 8738	. 2 class ⊥
2		vx	. . . . . 6 set x
3	2	cv 953	. . . . 5 class x
4		chil 8727	. . . . 5 class ℋ
5	3, 4	wss 2043	. . . 4 wff x ⊆ ℋ
6		vy	. . . . . 6 set y
7	6	cv 953	. . . . 5 class y
8		vz	. . . . . . . . . 10 set z
9	8	cv 953	. . . . . . . . 9 class z
10		vw	. . . . . . . . . 10 set w
11	10	cv 953	. . . . . . . . 9 class w
12		csp 8732	. . . . . . . . 9 class ·ih
13	9, 11, 12	co 3954	. . . . . . . 8 class (z ·ih w)
14		cc0 5214	. . . . . . . 8 class 0
15	13, 14	wceq 954	. . . . . . 7 wff (z ·ih w) = 0
16	15, 10, 3	wral 1642	. . . . . 6 wff ∀w ∈ x (z ·ih w) = 0
17	16, 8, 4	crab 1645	. . . . 5 class {z ∈ ℋ ∣∀w ∈ x (z ·ih w) = 0}
18	7, 17	wceq 954	. . . 4 wff y = {z ∈ ℋ ∣∀w ∈ x (z ·ih w) = 0}
19	5, 18	wa 223	. . 3 wff (x ⊆ ℋ ⋀ y = {z ∈ ℋ ∣∀w ∈ x (z ·ih w) = 0})
20	19, 2, 6	copab 2661	. 2 class {⟨x, y⟩∣(x ⊆ ℋ ⋀ y = {z ∈ ℋ ∣∀w ∈ x (z ·ih w) = 0})}
21	1, 20	wceq 954	1 wff ⊥ = {⟨x, y⟩∣(x ⊆ ℋ ⋀ y = {z ∈ ℋ ∣∀w ∈ x (z ·ih w) = 0})}

This definition is referenced by:  ocvalt 9092

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